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Den Hartog’s Mechanics

A web-based solutions manual for statics and dynamics

Problem 58

Let’s take the horizontal and vertical lines in the drawing in the book to be the x and y axes, respectively. The equation of the circular boundary, then, is

x^2 + y^2 = r^2

This is the radius of the thin disk shown in the drawing. The area of the upper and lower surfaces of the disk is \pi x^2 = \pi(r^2 - y^2), its thickness is dy, and its centroid is y from the flat side of the hemisphere. The center of gravity can be calculated through integration:

\bar y = \frac{\int_0^r \pi(r^2 - y^2) y\,dy}{\frac{2}{3}\pi r^3} = \frac{\frac{1}{4} \pi r^4}{\frac{2}{3}\pi r^3} = \frac{3}{8} r

Problem 59Problem 57

Last modified: January 22, 2009 at 8:32 PM.